Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms
Renaud Leplaideur Benoît Saussol
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 327-344 doi: 10.3934/dcds.2008.22.327
For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition.
keywords: thermodynamic formalism. large deviations return times
From local to global equilibrium states: Thermodynamic formalism via an inducing scheme
Renaud Leplaideur
Electronic Research Announcements 2014, 21(0): 72-79 doi: 10.3934/era.2014.21.72
We present a method to construct equilibrium states via inducing. This method can be used for some non-uniformly hyperbolic dynamical systems and for non-Hölder continuous potentials. It allows us to prove the existence of phase transition.
Fluctuations of the nth return time for Axiom A diffeomorphisms
Jean-René Chazottes Renaud Leplaideur
Discrete & Continuous Dynamical Systems - A 2005, 13(2): 399-411 doi: 10.3934/dcds.2005.13.399
We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition for an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure. As a by-product, we derive precise large deviation estimates and a central limit theorem for these return times. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a one-parameter family of induced transfer operators on unstable leaves crossing the visited set.
keywords: large deviations Successive return times Axiom A central limit theorem. transfer operator

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